3.402 \(\int \frac{1}{x^{5/2} \left (b x^2+c x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=350 \[ \frac{77 c^{9/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{30 b^{15/4} \sqrt{b x^2+c x^4}}-\frac{77 c^{9/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{15 b^{15/4} \sqrt{b x^2+c x^4}}+\frac{77 c^{5/2} x^{3/2} \left (b+c x^2\right )}{15 b^4 \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{77 c^2 \sqrt{b x^2+c x^4}}{15 b^4 x^{3/2}}+\frac{77 c \sqrt{b x^2+c x^4}}{45 b^3 x^{7/2}}-\frac{11 \sqrt{b x^2+c x^4}}{9 b^2 x^{11/2}}+\frac{1}{b x^{7/2} \sqrt{b x^2+c x^4}} \]

[Out]

1/(b*x^(7/2)*Sqrt[b*x^2 + c*x^4]) + (77*c^(5/2)*x^(3/2)*(b + c*x^2))/(15*b^4*(Sq
rt[b] + Sqrt[c]*x)*Sqrt[b*x^2 + c*x^4]) - (11*Sqrt[b*x^2 + c*x^4])/(9*b^2*x^(11/
2)) + (77*c*Sqrt[b*x^2 + c*x^4])/(45*b^3*x^(7/2)) - (77*c^2*Sqrt[b*x^2 + c*x^4])
/(15*b^4*x^(3/2)) - (77*c^(9/4)*x*(Sqrt[b] + Sqrt[c]*x)*Sqrt[(b + c*x^2)/(Sqrt[b
] + Sqrt[c]*x)^2]*EllipticE[2*ArcTan[(c^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(15*b^(15
/4)*Sqrt[b*x^2 + c*x^4]) + (77*c^(9/4)*x*(Sqrt[b] + Sqrt[c]*x)*Sqrt[(b + c*x^2)/
(Sqrt[b] + Sqrt[c]*x)^2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(3
0*b^(15/4)*Sqrt[b*x^2 + c*x^4])

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Rubi [A]  time = 0.860347, antiderivative size = 350, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{77 c^{9/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{30 b^{15/4} \sqrt{b x^2+c x^4}}-\frac{77 c^{9/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{15 b^{15/4} \sqrt{b x^2+c x^4}}+\frac{77 c^{5/2} x^{3/2} \left (b+c x^2\right )}{15 b^4 \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{77 c^2 \sqrt{b x^2+c x^4}}{15 b^4 x^{3/2}}+\frac{77 c \sqrt{b x^2+c x^4}}{45 b^3 x^{7/2}}-\frac{11 \sqrt{b x^2+c x^4}}{9 b^2 x^{11/2}}+\frac{1}{b x^{7/2} \sqrt{b x^2+c x^4}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(5/2)*(b*x^2 + c*x^4)^(3/2)),x]

[Out]

1/(b*x^(7/2)*Sqrt[b*x^2 + c*x^4]) + (77*c^(5/2)*x^(3/2)*(b + c*x^2))/(15*b^4*(Sq
rt[b] + Sqrt[c]*x)*Sqrt[b*x^2 + c*x^4]) - (11*Sqrt[b*x^2 + c*x^4])/(9*b^2*x^(11/
2)) + (77*c*Sqrt[b*x^2 + c*x^4])/(45*b^3*x^(7/2)) - (77*c^2*Sqrt[b*x^2 + c*x^4])
/(15*b^4*x^(3/2)) - (77*c^(9/4)*x*(Sqrt[b] + Sqrt[c]*x)*Sqrt[(b + c*x^2)/(Sqrt[b
] + Sqrt[c]*x)^2]*EllipticE[2*ArcTan[(c^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(15*b^(15
/4)*Sqrt[b*x^2 + c*x^4]) + (77*c^(9/4)*x*(Sqrt[b] + Sqrt[c]*x)*Sqrt[(b + c*x^2)/
(Sqrt[b] + Sqrt[c]*x)^2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(3
0*b^(15/4)*Sqrt[b*x^2 + c*x^4])

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Rubi in Sympy [A]  time = 83.225, size = 332, normalized size = 0.95 \[ \frac{1}{b x^{\frac{7}{2}} \sqrt{b x^{2} + c x^{4}}} - \frac{11 \sqrt{b x^{2} + c x^{4}}}{9 b^{2} x^{\frac{11}{2}}} + \frac{77 c \sqrt{b x^{2} + c x^{4}}}{45 b^{3} x^{\frac{7}{2}}} + \frac{77 c^{\frac{5}{2}} \sqrt{b x^{2} + c x^{4}}}{15 b^{4} \sqrt{x} \left (\sqrt{b} + \sqrt{c} x\right )} - \frac{77 c^{2} \sqrt{b x^{2} + c x^{4}}}{15 b^{4} x^{\frac{3}{2}}} - \frac{77 c^{\frac{9}{4}} \sqrt{\frac{b + c x^{2}}{\left (\sqrt{b} + \sqrt{c} x\right )^{2}}} \left (\sqrt{b} + \sqrt{c} x\right ) \sqrt{b x^{2} + c x^{4}} E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{15 b^{\frac{15}{4}} x \left (b + c x^{2}\right )} + \frac{77 c^{\frac{9}{4}} \sqrt{\frac{b + c x^{2}}{\left (\sqrt{b} + \sqrt{c} x\right )^{2}}} \left (\sqrt{b} + \sqrt{c} x\right ) \sqrt{b x^{2} + c x^{4}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{30 b^{\frac{15}{4}} x \left (b + c x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**(5/2)/(c*x**4+b*x**2)**(3/2),x)

[Out]

1/(b*x**(7/2)*sqrt(b*x**2 + c*x**4)) - 11*sqrt(b*x**2 + c*x**4)/(9*b**2*x**(11/2
)) + 77*c*sqrt(b*x**2 + c*x**4)/(45*b**3*x**(7/2)) + 77*c**(5/2)*sqrt(b*x**2 + c
*x**4)/(15*b**4*sqrt(x)*(sqrt(b) + sqrt(c)*x)) - 77*c**2*sqrt(b*x**2 + c*x**4)/(
15*b**4*x**(3/2)) - 77*c**(9/4)*sqrt((b + c*x**2)/(sqrt(b) + sqrt(c)*x)**2)*(sqr
t(b) + sqrt(c)*x)*sqrt(b*x**2 + c*x**4)*elliptic_e(2*atan(c**(1/4)*sqrt(x)/b**(1
/4)), 1/2)/(15*b**(15/4)*x*(b + c*x**2)) + 77*c**(9/4)*sqrt((b + c*x**2)/(sqrt(b
) + sqrt(c)*x)**2)*(sqrt(b) + sqrt(c)*x)*sqrt(b*x**2 + c*x**4)*elliptic_f(2*atan
(c**(1/4)*sqrt(x)/b**(1/4)), 1/2)/(30*b**(15/4)*x*(b + c*x**2))

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Mathematica [C]  time = 0.23819, size = 210, normalized size = 0.6 \[ \frac{-\sqrt{\frac{i \sqrt{c} x}{\sqrt{b}}} \left (10 b^3-22 b^2 c x^2+154 b c^2 x^4+231 c^3 x^6\right )-231 \sqrt{b} c^{5/2} x^5 \sqrt{\frac{c x^2}{b}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c} x}{\sqrt{b}}}\right )\right |-1\right )+231 \sqrt{b} c^{5/2} x^5 \sqrt{\frac{c x^2}{b}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c} x}{\sqrt{b}}}\right )\right |-1\right )}{45 b^4 x^{7/2} \sqrt{\frac{i \sqrt{c} x}{\sqrt{b}}} \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^(5/2)*(b*x^2 + c*x^4)^(3/2)),x]

[Out]

(-(Sqrt[(I*Sqrt[c]*x)/Sqrt[b]]*(10*b^3 - 22*b^2*c*x^2 + 154*b*c^2*x^4 + 231*c^3*
x^6)) + 231*Sqrt[b]*c^(5/2)*x^5*Sqrt[1 + (c*x^2)/b]*EllipticE[I*ArcSinh[Sqrt[(I*
Sqrt[c]*x)/Sqrt[b]]], -1] - 231*Sqrt[b]*c^(5/2)*x^5*Sqrt[1 + (c*x^2)/b]*Elliptic
F[I*ArcSinh[Sqrt[(I*Sqrt[c]*x)/Sqrt[b]]], -1])/(45*b^4*x^(7/2)*Sqrt[(I*Sqrt[c]*x
)/Sqrt[b]]*Sqrt[x^2*(b + c*x^2)])

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Maple [A]  time = 0.03, size = 237, normalized size = 0.7 \[{\frac{c{x}^{2}+b}{90\,{b}^{4}} \left ( 462\,\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) \sqrt{2}{x}^{4}b{c}^{2}-231\,\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) \sqrt{2}{x}^{4}b{c}^{2}-462\,{c}^{3}{x}^{6}-308\,b{c}^{2}{x}^{4}+44\,{b}^{2}c{x}^{2}-20\,{b}^{3} \right ){x}^{-{\frac{3}{2}}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(5/2)/(c*x^4+b*x^2)^(3/2),x)

[Out]

1/90/(c*x^4+b*x^2)^(3/2)/x^(3/2)*(c*x^2+b)*(462*((c*x+(-b*c)^(1/2))/(-b*c)^(1/2)
)^(1/2)*((-c*x+(-b*c)^(1/2))/(-b*c)^(1/2))^(1/2)*(-x*c/(-b*c)^(1/2))^(1/2)*Ellip
ticE(((c*x+(-b*c)^(1/2))/(-b*c)^(1/2))^(1/2),1/2*2^(1/2))*2^(1/2)*x^4*b*c^2-231*
((c*x+(-b*c)^(1/2))/(-b*c)^(1/2))^(1/2)*((-c*x+(-b*c)^(1/2))/(-b*c)^(1/2))^(1/2)
*(-x*c/(-b*c)^(1/2))^(1/2)*EllipticF(((c*x+(-b*c)^(1/2))/(-b*c)^(1/2))^(1/2),1/2
*2^(1/2))*2^(1/2)*x^4*b*c^2-462*c^3*x^6-308*b*c^2*x^4+44*b^2*c*x^2-20*b^3)/b^4

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}} x^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + b*x^2)^(3/2)*x^(5/2)),x, algorithm="maxima")

[Out]

integrate(1/((c*x^4 + b*x^2)^(3/2)*x^(5/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (c x^{6} + b x^{4}\right )} \sqrt{c x^{4} + b x^{2}} \sqrt{x}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + b*x^2)^(3/2)*x^(5/2)),x, algorithm="fricas")

[Out]

integral(1/((c*x^6 + b*x^4)*sqrt(c*x^4 + b*x^2)*sqrt(x)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{\frac{5}{2}} \left (x^{2} \left (b + c x^{2}\right )\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**(5/2)/(c*x**4+b*x**2)**(3/2),x)

[Out]

Integral(1/(x**(5/2)*(x**2*(b + c*x**2))**(3/2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}} x^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + b*x^2)^(3/2)*x^(5/2)),x, algorithm="giac")

[Out]

integrate(1/((c*x^4 + b*x^2)^(3/2)*x^(5/2)), x)